函數(shù)英文知識課件

FunctionEnglishPPTcoursewarecontents目錄DefinitionandpropertiesoffunctionsCommonfunctiontypesOperationandTransformationofFunctionsApplicationoffunctionsTheLimitandContinuityofFunctions01DefinitionandpropertiesoffunctionsAfunctionisafundamentalconceptinmathematics,whichisamappingrelationshipfromaninputsettoanoutputset.Thefunctiondefinitionusuallyincludesasetofinputvalues,anoutputvalueset,andacorrespondingrule,thatis,theinputvalueobtainstheoutputvaluethroughacertainoperationalrelationship.DefinitionoffunctionsThepropertiesoffunctionsincludedomain,range,monotonicity,parity,andsoon.Adomainisasetofinputvaluesforafunction,andavaluedomainisasetofoutputvaluesforafunction.Monotonicitydescribestheincreaseordecreaseofafunctionwithinacertaininterval,whileparitydescribeswhetherthefunctionissymmetricabouttheorigin.ThepropertiesoffunctionsFunctionscanbeclassifiedaccordingtodifferentcriteria,suchaslinearfunctions,quadraticfunctions,trigonometricfunctions,etc.Functionscanalsobedividedintoopenfunctions,closedfunctions,semiopenandsemiclosedfunctionsbasedontherelationshipbetweentheirdomainofdefinitionandrangeofvalues.Inaddition,functionscanalsobedividedintodifferentiableandnondifferentiablefunctionsbasedonwhethertheyaredifferentiable.Classificationoffunctions02CommonfunctiontypesLinearfunctionisacommontypeoffunctioninmathematics,representedbyastraightline.Thegeneralformofalinearfunctionisy=ax+b,whereaandbareconstantsanda≠0.Thecharacteristicsoflinearfunctionsinclude:whena>0,thefunctionisanincreasingfunction;Whena<0,thefunctionisasubtractionfunction.Linearfunctionshaveawiderangeofapplicationsinreallife,suchaseconomics,statistics,andotherfields.LinearfunctionQuadraticfunctionisacommontypeofmathematicalfunction,withageneralformofy=ax^2+bx+c,wherea,b,andcareconstantsanda≠0.QuadraticfunctionThegraphofaquadraticfunctionisaparabolawithvertexcoordinates(-b/2a,c-b^2/4a).Thecharacteristicsofquadraticfunctionsinclude:whena>0,theopeningoftheparabolaisupward;Whena<0,theopeningoftheparabolaisdownward.Quadraticfunctionshaveawiderangeofapplicationsinreallife,suchasinphysics,engineering,andotherfields.QuadraticfunctionTrigonometricfunctionTrigonometricfunctionsareatypeofmathematicalfunctionthatmainlyincludessinefunctions,cosinefunctions,andtangentfunctions.Thedomainoftrigonometricfunctionsisanintervalontherealnumberaxis,andtherangeofvaluesisthe[-1,1]interval.Thecharacteristicsoftrigonometricfunctionsinclude:bothsineandcosinefunctionsareperiodicfunctionswithperiodicity;Thetangentfunctionisanoddfunctionwithparity.Trigonometricfunctionsarealsowidelyusedinreallife,suchasinfieldssuchasphysicsandengineering.Exponentialfunctionsandnaturallogarithmicfunctionsaretwocommontypesofmathematicalfunctions.Thegeneralformofanexponentialfunctionisy=a^x(a>0,a≠1),whilethegeneralformofalogarithmicfunctionisy=log_Ax(a>0,a≠1).Thecharacteristicsofexponentialfunctionsinclude:whena>1,thefunctionisanincreasingfunction;When0<a<1,thefunctionisasubtractionfunction.Thecharacteristicsoflogarithmicfunctionsinclude:logarithmicfunctionsaremonotonicallyincreasingwithintheirdomainofdefinition.Exponentialandlogarithmicfunctionsarealsowidelyusedinreallife,suchasinfieldssuchasstatisticsandfinance.Exponentialandlogisticfunctions03OperationandTransformationofFunctionsFouroperationsoffunctionsAdditionoffunctions:AddingtwofunctionsmeanstoaddtheircorrespondingoutputsforthesameinputForexample,iff(x)=x^2andg(x)=3x+2,thenf(x)+g(x)=(x^2+3x+2)forallxSubtractionoffunctions:Subtractingtwofunctionsissimilartoaddition,butwithnegativenumbersIff(x)=x^2andg(x)=3x+2,thenf(x)-g(x)=(x^2-3x-2)forallxMultiplicationoffunctions:MultiplyingtwofunctionsmeanstomultiplytheircorrespondingoutputsforthesameinputForexample,iff(x)=x^2andg(x)=3x+2,thenf(x)*g(x)=(3x^3+2x^2)forallxDivisionoffunctions:Dividingonefunctionbyanotherismorecomplex,butcanbedonebyfindingtherecipeofthesecondfunctionandthenmultiplyingthefirstfunctionbyit.Forexample,iff(x)=x^2andg(x)=3x+2,thenf(x)/g(x)=(x^2/(3x+2))forallxexception-2/3CompositeoperationoffunctionsDefinition:Thecompositefunctionf(g(x))isformedbyreplacingtheinputoffunctiongwiththeoutputoffunctionf.Forexample,iff(x)=x^2andg(x)=3x+2,thenthecompositefunctionf(g(x))=(3x+2)^2forallxProperties:ThecompositefunctionhasthepropertiesofboththeinnerandouterfunctionsForexample,iff(g(x))=(3x+2)^2andh(x)=x+1,then(f\circleg)(h(x))=(3(x+1)+2)^2)=(3x+5)^2)Inverses:Ifafunctionhasaninverse,itcanbeusedto"undo"thefunctionForexample,theinverseoff(x)=x^2isf^(-1)(x)=sqrt(x),whichexplainsthesquaringoperationCompositionrules:Whencomposingmultiplefunctions,theorderinwhichtheyarecomposedmaterialsForexample,iff(g(h(x))),thentheorderisimportantbecausehwillbeappliedfirst,theng,thenfHorizontalshiftShiftingafunctionHorizontallyisequivalenttoaddingorsubtractingaconstantfromallinputsForexample,iff(x)=x^2isshiftedleftby1unit,itbenefits(x-1)^2VerticalshiftShiftingafunctionverticallyisequivalenttoaddingorsubtractingaconstantfromalloutputsForexample,iff(x)=x^2isshiftedupby1unit,itbenefitsy=x^2+1ReflectionReflectingafunctionacrossthexoryaxiscreatesanewfunctionwiththesameinputsbutoppositeoutputsForexample,iff(x)=x^2isreflectedacrosstheyaxis,itbenefitsy=-(x^2)Imagetransformationoffunctions04ApplicationoffunctionsTrigonometricFunctionsDescribeperiodicmotion,soundwaves,andothernaturalphenomenaCalculusFunctionsUsedinfindingareas,volumes,andoptimizingprocessesAlgebraicFunctionsUsedtosolvevariousAlgebraicequationsandreportrelationshipsbetweenvariablesTheApplicationofFunctionsinMathematics

TheApplicationofFunctionsinRealLifeFinancialFunctionsUsedincalculatinginterest,loans,andotherfinancialtransactionsEngineeringFunctionsDescribephysicalpropertiesofmaterialsandsystemsindifferentfieldsStatisticalFunctionsDescribepatternsandrelationshipsindatacollectedfromexperimentsorsurveys123UsedinfindingsolutionstocomplexmathematicalproblemsNumericalAnalysisCreatemodelstoreportrealworldsystemsandpredicttheirbehaviorSimulationandModelingExtractmeaningfulinformationfromlargedatasetstounderstandpatternsandtrendsDataAnalysisTheApplicationofFunctionsinScientificComputing05TheLimitandContinuityofFunctionsDefinition01Thelimitofafunctionf(x)asxapproachesaisLifforallpositivenumbersepsilon,thereexistsapositivenumberdeltasothatif0<|x-a|<delta,then0<|f(x)-L|<epsilonProperties02Limitlawssuchasaddition,subtraction,multiplication,division,andexpansioncanbeappliedtolimitsApplications03Limitsareusedinthestudyofcalculus,analysis,andrealanalysistostudythebehavioroffunctionsnearspecificpointsorinspecificintervalsLimitoffunctionDefinitionAfunctionfiscontinuousatapointaifthelimitoff(x)asxapproachesaisequaltof(a)PropertiesIfafunctioniscontinuousatapoint,thenitis